The solution of the linear fractional partial differential equations using the homotopy analysis method

Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

Volumn 65, Issue 11, 2010, Pages 935-949

  Dehghan, Mehdi ^a   Manafian, Jalil ^a   Saadatmandi, Abbas ^b  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b UNIVERSITY OF KASHAN   (Iran)

Author keywords

Analytical Solution; Fractional Partial Differential Equations (FPDEs); Homotopy Analysis Method (HAM)

Indexed keywords

EID: 78649487054     PISSN: 09320784     EISSN: None     Source Type: Journal    
DOI: 10.1515/zna-2010-1106     Document Type: Article

References (70)

1. B. J. West, M. Bolognab, and P. Grigolini, Physics of fractal operators, Springer, New York 2003.
2. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York 1993.
3. I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, New York 1999.
4. S.G. Samko, A.A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, Amsterdam 1993.
5. S. Abbasbandy, J. Comput. Appl. Math. 207, 53 (2007).
6. G. Adomian, J. Math. Anal. Appl. 55, 441 (1976).
7. S. Momani and N. T. Shawagfeh, Appl. Math. Comput. 182, 1083 (2006).
8. S. Kemple and H. Beyer, Global and causal solutions of fractional differential equations, Transform methods and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP), Singapore, 96, 210 (1997).
9. L. Debnath and D. Bhatta, Integral transforms and their applications (second edition), Chapman and Hall/CRC, Boca Raton 2007.
10. K. Al-Khaled and S. Momani, Appl. Math. Comput. 165, 473 (2005).
11. M. Caputo, J. R. Astron. Soc. 13, 529 (1967).
12. L. Debnath, Int. J. Math. Math. Sci. 54, 3413 (2003).
13. K. Diethelm, N. J. Ford, and A.D. Freed, Nonlinear Dyn. 29, 3 (2002).
14. H. Jafari and S. Seifi, Commun. Nonlinear Sci. Numer. Simul. 14, 2006 (2009).
15. H. Jafari and S. Seifi, Commun. Nonlinear Sci. Numer. Simul. 14, 1962 (2009).
16. T. Hayat, S. Nadeem, and S. Asghar, Appl.Math. Comput. 151, 153 (2004).
17. M. Khan and T. Hayat, Nonlinear Anal.: Real World Appl. 9, 1952 (2008).
18. A. A. Kilbas and J. J. Trujillo, Appl. Anal. 78, 153 (2001).
19. S.V. Kiryakova, Appl.Math. Comput. 118, 441 (2000).
20. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York 1974.
21. S. S. Ray and R.K. Bera, Appl. Math. Comput. 167, 561 (2005).
22. N. T. Shawagfeh, Appl. Math. Comput. 131, 241 (2002).
23. L. Song and H. Zhang, Chaos, Solitons, and Fractals 40, 1616 (2009).
24. L. Song and H. Zhang, Phys. Let. A 367, 88 (2007).
25. H. Xu and C. Jie, Phys. Let. A 372, 1250 (2008).
26. Z. Odibat, S. Momani, and H. Xu, Appl. Math. Model. 34, 593 (2010).
27. M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, Appl. Math. Model. 34, 24 (2010).
28. Z. Odibat and S. Momani, Appl. Math. Model. 32, 28 (2008).
29. S. Abbasbandy and F. Samadian Zakaria, Nonlinear Dyn. 51, 83 (2008).
30. S. Abbasbandy, Z. Angew. Math. Phys. 59, 51 (2008).
31. S. Abbasbandy, Phys. Lett. A 361, 478 (2007).
32. T. Hayat, M. Khan, and M. Ayub, Int. J. Eng. Sci. 42, 123 (2004).
33. T. Hayat, M. Khan, and S. Asghar, Appl. Math. Comput. 155, 417 (2004).
34. T. Hayat, M. Khan, and M. Ayub, J. Math. Anal. Appl. 298, 225 (2004).
35. T. Hayat, M. Khan, and M. Ayub, Z. Angew. Math. Phys. 56, 1012 (2005).
36. T. Hayat and M. Khan, Nonlinear Dyn. 42, 395 (2005).
37. A. Saadatmandi and M. Dehghan, Comput. Math. Appl. 59, 1326 (2010).
38. M. Khan and S. Wang, Nonlinear Anal.: Real World Appl. 10, 203 (2009).
39. M. Khan, S. Hyder Ali, and H. Qi, Nonlinear Anal.: Real World Appl. 10, 980 (2009).
40. M. Khan, T. Hayat, and S. Asghar, Int. J. Eng. Sci. 44, 333 (2006).
41. M. Khan, K. Maqbool, and T. Hayat, Acta Mech. 184, 1 (2006).
42. M. Khan, J. Porous Med. 10, 473 (2007).
43. M. Khan, J. Porous Med. 12, 919 (2009).
44. S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University 1992.
45. S. J. Liao, Stud. Appl. Math. 117, 2529 (2006).
46. S. J. Liao, Int. J. Nonlinear Mech. 30, 371 (1995).
47. S. J. Liao, Int. J. Nonlinear Mech. 32, 815 (1997).
48. S. J. Liao, Int. J. Nonlinear Mech. 34, 759 (1999).
49. S. J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, Chapman and Hall, CRC Press, Boca Raton 2003.
50. S. J. Liao, Appl. Math. Comput. 147, 499 (2004).
51. M. S.H. Chowdhury, I. Hashim, and O. Abdulaziz, Commun. Nonlinear Sci. Numer. Simul. 14, 371 (2009).
52. G. Domairry and M. Fazeli, Commun. Nonlinear Sci. Numer. Simul. 14, 489 (2009).
53. G. Domairry and N. Nadim, Int. Commun. Heat Mass Transf. 35, 93 (2008).
54. M. Sajid and T. Hayat, Nonlinear Anal.: Real World Appl. 9, 2296 (2008).
55. A. Fakhari and G. Domairry, Phys. Let. A 368, 64 (2007).
56. T. Hayat and M. Sajid, Int. J. Eng. Sci. 45, 393 (2007).
57. M. Sajid and T. Hayat, Chaos, Solitons, and Fractals 38, 506 (2008).
58. M. Dehghan, J. Manafian, and A. Saadatmandi, Numer. Methods Part. Diff. Eqs. J. 26, 448 (2010).
59. H. Xu and S. J. Liao, J. Non-Newton. Fluid Mech. 129, 46 (2005).
60. S. Abbasbandy, Chem. Eng. J. 136, 144 (2008).
61. F. Shakeri and M. Dehghan, Math. Comput.Model. 48, 486 (2008).
62. M. Dehghan and F. Shakeri, Numer. Method Part. Diff. Eqs. 25, 1238 (2009).
63. A. Saadatmandi, M. Dehghan, and A. Eftekhari, Nonlinear Anal.: Real World Appl. 10, 1912 (2009).
64. M. Tatari and M. Dehghan, Comput. Math. Appl. 58, 2160 (2009).
65. M. Tatari and M. Dehghan, J. Comput. Appl. Math. 207, 121 (2007).
66. M. Dehghan and R. Salehi, Commun. Numer. Methods Eng., DOI: 10.1002/cnm.1315.
67. M. Dehghan, M. Shakourifar, and A. Hamidi, Chaos, Solitons, and Fractals 39, 2509 (2009).
68. M. Dehghan and F. Shakeri, Physica Scripta 78, 1 (2008).
69. F. Shakeri and M. Dehghan, Z. Naturforsch. 65a, 453 (2010).
70. M. Dehghan and R. Salehi, Comput. Phys. Commun. 181, 1255 (2010).